A005361 -id:A005361 - cp's OEIS frontend

A005934 Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization (A005361).

1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1296, 1728, 2592, 3456, 5184, 7776, 10368, 15552, 20736, 31104, 41472, 62208, 86400, 108000, 129600, 194400, 216000, 259200, 324000, 432000, 518400, 648000, 972000, 1296000, 1944000, 2592000

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..609 (terms 1..300 from T. D. Noe)
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991.
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)
C. B. Lacampagne and J. L. Selfridge, Large highly powerful numbers are cubeful, Proc. Amer. Math. Soc., Vol. 91, No. 2 (1984), pp. 173-181.
Wikipedia, Highly powerful number.
Index entries for sequences related to powerful numbers

Cf. A036965, A001694, A007532, A005361, A005188, A003321, A014576, A023052, A046074.

Cf. A085629.

a = {1}; b = {1}; f[n_] := Times @@ Last /@ FactorInteger[n]; Do[If[f@ n > Max[b], And[AppendTo[b, f@ n], AppendTo[a, n]]], {n, 1000000}]; a (* Michael De Vlieger, Aug 28 2015 *)
With[{s = Array[Times @@ FactorInteger[#][[All, -1]] &, 3*10^6]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Oct 15 2017 *)
DeleteDuplicates[Table[{n,Times@@FactorInteger[n][[All,2]]},{n,26*10^5}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]] (* Harvey P. Dale, May 13 2022 *)

{prdex(n)=local(s,fac); s=1; fac=factor(n); for(k=1,matsize(fac)[1],s=s*fac[k,2]); return(s)} {hp(m)=local(rec); rec=0; for(n=1,m,if(prdex(n)>rec,rec=prdex(n); print1(n",")))}

For n = Product p_i^e_i, let b(n) = Product e_i; then n is highly powerful if b(n) sets a new record.

Hardy and Subbarao give an extensive table.

Corrected and extended by Jason Earls, Jul 10 2003

A322327 a(n) = A005361(n) * A034444(n).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 6, 4, 4, 2, 8, 2, 4, 4, 8, 2, 8, 2, 8, 4, 4, 2, 12, 4, 4, 6, 8, 2, 8, 2, 10, 4, 4, 4, 16, 2, 4, 4, 12, 2, 8, 2, 8, 8, 4, 2, 16, 4, 8, 4, 8, 2, 12, 4, 12, 4, 4, 2, 16, 2, 4, 8, 12, 4, 8, 2, 8, 4, 8, 2, 24, 2, 4, 8, 8, 4, 8, 2, 16, 8, 4, 2, 16, 4, 4, 4, 12, 2, 16, 4, 8, 4, 4, 4, 20, 2, 8, 8, 16

Offset: 1

Werner Schulte, Dec 03 2018

Conjecture: Let k be some fixed integer and a_k(n) = A005361(n) * k^A001221(n) for n > 0 with 0^0 = 1. Then a_k(n) is multiplicative with a_k(p^e) = k*e for prime p and e > 0. For k = 0 see A000007 (offset 1), for k = 1 see A005361, for k = 2 see this sequence, for k = 3 see A226602 (offset 1), and for k = 4 see A322328.

Dirichlet inverse b(n) [= A355837(n)] is multiplicative with b(p^e) = 2 * (e mod 2) * (-1)^((e+1)/2) for prime p and e > 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A000007, A001221, A001620, A005361, A034444, A226602, A227291, A286324, A322328, A355837 (Dirichlet inverse).

a[n_] := If[n==1, 1, Module[{f = FactorInteger[n]}, 2^Length[f] * Times@@f[[;;,2]]]]; Array[a, 100] (* Amiram Eldar, Dec 03 2018 *)

a(n) = my(f=factor(n)); vecprod(f[,2])*2^omega(n); \\ Michel Marcus, Dec 04 2018

A322327(n) = factorback(apply(e -> e+e, factor(n)[, 2])); \\ Antti Karttunen, Jul 18 2022

from math import prod
from sympy import factorint
def A322327(n): return prod(e<<1 for e in factorint(n).values()) # Chai Wah Wu, Dec 26 2022

Multiplicative with a(p^e) = 2*e for prime p and e > 0.
Dirichlet g. f.: (zeta(s))^2 * zeta(2*s) / zeta(4*s).
Equals Dirichlet convolution of A000005 and A227291.
Sum_{k=1..n} a(k) ~ 15*(log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 - 360*zeta'(4)/Pi^4) * n / Pi^2 + 6*zeta(1/2)^2 * sqrt(n) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 20 2020
a(n) = A005361(n^2) = A286324(n^2). - Amiram Eldar, Dec 09 2023

Data section extended up to a(100) by Antti Karttunen, Jul 18 2022

A284001 a(n) = A005361(A283477(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 4, 6, 1, 2, 4, 6, 8, 12, 18, 24, 1, 2, 4, 6, 8, 12, 18, 24, 16, 24, 36, 48, 54, 72, 96, 120, 1, 2, 4, 6, 8, 12, 18, 24, 16, 24, 36, 48, 54, 72, 96, 120, 32, 48, 72, 96, 108, 144, 192, 240, 162, 216, 288, 360, 384, 480, 600, 720, 1, 2, 4, 6, 8, 12, 18, 24, 16, 24, 36, 48, 54, 72, 96, 120, 32, 48, 72, 96, 108, 144, 192, 240, 162, 216, 288, 360, 384, 480

Offset: 0

Antti Karttunen, Mar 18 2017

a(n) is the product of elements of the multiset that covers an initial interval of positive integers with multiplicities equal to the parts of the n-th composition in standard order (graded reverse-lexicographic, A066099). This composition is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. For example, the 13th composition is (1,2,1) giving the multiset {1,2,2,3} with product 12, so a(13) = 12. - Gus Wiseman, Apr 26 2020

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A005361, A283477, A284002, A284005.
Row products of A095684.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Weighted sum is A029931.
- Necklaces are A065609.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Distinct parts are counted by A334028.
Cf. A003963, A034691, A053645, A057335, A305936, A318284, A333764, A333940, A333942, A334030.

Table[Times @@ FactorInteger[#][[All, -1]] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]]], {n, 0, 93}] (* Michael De Vlieger, Mar 18 2017 *)

A005361(n) = factorback(factor(n)[, 2]); \\ From A005361
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A283477(n) = A108951(A019565(n));
A284001(n) = A005361(A283477(n));

(define (A284001 n) (A005361 (A283477 n)))

a(n) = A005361(A283477(n)).
a(n) = A003963(A057335(n)). - Gus Wiseman, Apr 26 2020
a(n) = A284005(A053645(n)) for n > 0 with a(0) = 1. - Mikhail Kurkov, Jun 05 2021 [verification needed]

A322328 a(n) = A005361(n) * 4^A001221(n) for n > 0.

Original entry on oeis.org

1, 4, 4, 8, 4, 16, 4, 12, 8, 16, 4, 32, 4, 16, 16, 16, 4, 32, 4, 32, 16, 16, 4, 48, 8, 16, 12, 32, 4, 64, 4, 20, 16, 16, 16, 64, 4, 16, 16, 48, 4, 64, 4, 32, 32, 16, 4, 64, 8, 32, 16, 32, 4, 48, 16, 48, 16, 16, 4, 128, 4, 16, 32, 24, 16, 64, 4, 32, 16, 64, 4

Offset: 1

Werner Schulte, Dec 03 2018

Let k be some fixed integer and a_k(n) = A005361(n) * k^A001221(n) for n > 0 with 0^0 = 1. Then a_k(n) is multiplicative with a_k(p^e) = k*e for prime p and e > 0. For k = 0 see A000007 (offset 1), for k = 1 see A005361, for k = 2 see A322327, for k = 3 see A226602 (offset 1), and for k = 4 see this sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000007, A001221, A005361, A007427, A008836, A034444, A226602, A322327.

f:= n -> mul(4*t[2],t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Dec 07 2018

a[n_] := If[n==1, 1, Module[{f = FactorInteger[n]}, 4^Length[f] * Times@@f[[;; , 2]]]]; Array[a, 100] (* Amiram Eldar, Dec 03 2018 *)

a(n) = my(f=factor(n)); vecprod(f[,2])*4^omega(n); \\ Michel Marcus, Dec 04 2018

from math import prod
from sympy import factorint
def A322328(n): return prod(e<<2 for e in factorint(n).values()) # Chai Wah Wu, Dec 24 2022

Multiplicative with a(p^e) = 4*e for prime p and e > 0.
Dirichlet g.f.: (zeta(s))^4 / (zeta(2*s))^2.
Dirichlet inverse is b(n) = a(n) * A008836(n) for n > 0, and b(n) is multiplicative with b(p^e) = 4*e*(-1)^e for prime p and e > 0.
Equals Dirichlet convolution of A034444 with itself.
Equals Dirichlet convolution of A000005 with abs(A007427).

A353500 Numbers that are the smallest number with product of prime exponents k for some k. Sorted positions of first appearances in A005361, unsorted version A085629.

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1152, 1296, 1728, 2048, 2592, 3456, 5184, 7776, 8192, 10368, 13824, 15552, 18432, 20736, 31104, 41472, 55296, 62208, 73728, 86400, 108000, 129600, 131072, 165888, 194400, 216000, 221184, 259200, 279936, 324000

Offset: 1

Gus Wiseman, May 17 2022

nonn

All terms are highly powerful (A005934), but that sequence looks only at first appearances that reach a record, and is missing 1152, 2048, 8192, etc.

			The prime exponents of 86400 are (7,3,2), and this is the first case of product 42, so 86400 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Highly powerful number.

These are the positions of first appearances in A005361, counted by A266477.
This is the sorted version of A085629.
The version for shadows instead of exponents is A353397, firsts in A353394.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices, counted by A339095.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime exponents, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
Cf. A070175, A097318, A116608, A162247, A182850, A304678, A325131, A325238, A353399, A353503, A353506, A353507.
Subsequence of A181800.

nn=1000;
d=Table[Times@@Last/@FactorInteger[n],{n,nn}];
Select[Range[nn],!MemberQ[Take[d,#-1],d[[#]]]&]
lps[fct_] := Module[{nf = Length[fct]}, Times @@ (Prime[Range[nf]]^Reverse[fct])]; lps[{1}] = 1; q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, (n == 1 || AllTrue[e, # > 1 &]) && n == Min[lps /@ f[Times @@ e]]]; Select[Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]], q] (* Amiram Eldar, Sep 29 2024, using the function f by T. D. Noe at A162247 *)

A085651 Index of the first of two successive 2's in A005361.

Original entry on oeis.org

44, 49, 75, 98, 116, 147, 171, 244, 260, 275, 315, 332, 363, 387, 475, 476, 507, 524, 531, 548, 549, 603, 604, 636, 692, 724, 725, 747, 764, 774, 819, 844, 845, 846, 867, 908, 924, 931, 963, 1035, 1075, 1083, 1179, 1196, 1251, 1274, 1275, 1324, 1340, 1395

Offset: 1

Amarnath Murthy, Jason Earls, Jul 11 2003

Numbers such that bigomega(n) - omega(n) = 1 and bigomega(n+1) - omega(n+1) = 1, where bigomega(n) is the number of primes dividing n (counted with repetition) and omega(n) is the number of distinct primes dividing n. - Michel Lagneau, Dec 17 2011
Elements of A060687 whose successor is also in A060687. - Emmanuel Vantieghem, Mar 05 2017
This sequence has 3548 terms up to 10^5, 35340 up to 10^6, 353147 up to 10^7, and 3531738 up to 10^8, suggesting a natural density around 0.0353.... - Charles R Greathouse IV, Mar 06 2017

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Indranil Ghosh)
Indranil Ghosh, Python program to generate the sequence.

Cf. A005361, A060687.

Select[Range[1400],PrimeOmega[#]-PrimeNu[#] == 1 && PrimeOmega[#+1] - PrimeNu[#+1] == 1 &] (* Indranil Ghosh, Mar 05 2017 *)
SequencePosition[Table[PrimeOmega[n]-PrimeNu[n],{n,1500}],{1,1}][[;;,1]] (* Harvey P. Dale, Jul 17 2024 *)

isok(n) = (bigomega(n)-omega(n) == 1) && (bigomega(n+1)-omega(n+1) == 1); \\ Michel Marcus, Mar 05 2017

is(n)=factorback(factor(n)[,2])==2 && factorback(factor(n+1)[,2])==2 \\ Charles R Greathouse IV, Mar 06 2017

A052306 Product of exponents of prime factorization of n by prime signature: A005361(A025487).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 3, 1, 5, 4, 4, 2, 6, 6, 5, 3, 7, 8, 4, 6, 1, 9, 4, 8, 10, 6, 7, 2, 12, 5, 9, 12, 8, 8, 3, 15, 8, 6, 10, 9, 14, 4, 16, 10, 9, 4, 18, 12, 7, 11, 12, 16, 1, 6, 20, 12, 10, 5, 21, 16, 8, 12, 15, 18, 2, 8, 24, 18, 14, 11, 8, 16, 6, 24, 20, 9, 9, 25, 13, 18, 20, 3

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

Álvar Ibeas, Table of n, a(n) for n = 1..10000

Cf. A005361, A025487.

lista() = {v = readvec("b025487.txt"); for (i=1, #v, f = factor(v[i]); print1(prod(k=1, #f~, f[k,2]), ", "););} \\ Michel Marcus, Nov 02 2014

Offset updated by Matthew Vandermast, Nov 08 2008

A082091 a(n) = one more than the number of iterations of A005361 needed to reach 1 from the starting value n.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 4, 3, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 4, 4, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 3, 4, 2, 2, 2, 3, 2, 2

Offset: 1

Labos Elemer, Apr 09 2003

nonn

			For n = 2 = 2^1, A005361(2) = 1, so we reach 1 in one step, and thus a(2) = 1+1 = 2.
For n = 4 = 2^2, A005361(4) = 2; A005361(2) = 1, so we reach 1 in two steps, and thus a(4) = 2+1 = 3.
For n = 6 = 2^1 * 3^1, A005361(6) = 1*1 = 1, so we reach 1 in one step, and thus a(6) = 1+1 = 2.
For n = 64 = 2^6, A005361(64) = 6, thus a(64) = 1 + a(6) = 3.
For n = 10! = 3628800 = 2^8 * 3^4 * 5^2 * 7*1, A005361(3628800) = 64, thus a(3628800) = 1 + a(64) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A001414, A005361, A008475, A056239, A082083-A082086, A082090.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] expr[x_] := Apply[Times, ep[x]] Table[Length[FixedPointList[expr, w]]-1, {w, 2, 128}]
(* Second program: *)
Table[Length@ NestWhileList[Apply[Times, FactorInteger[#][[All, -1]]] &, n, # != 1 &], {n, 105}] (* Michael De Vlieger, Jul 29 2017 *)

A005361(n) = factorback(factor(n)[, 2]); \\ This function from Charles R Greathouse IV, Nov 07 2014
A082091(n) = if(1==n,1,1+A082091(A005361(n))); \\ Antti Karttunen, Jul 28 2017

first(n) = my(v = vector(n)); v[1] = 1; for(i=2, n, v[i] = v[factorback(factor(i)[, 2])] + 1); v \\ David A. Corneth, Jul 28 2017

(define (A082091 n) (if (= 1 n) n (+ 1 (A082091 (A005361 n))))) ;; Antti Karttunen, Jul 28 2017

a(1) = 1, and for n > 1, a(n) = 1 + a(A005361(n)).

Term a(1)=1 prepended, Name and Example sections edited by Antti Karttunen, Jul 28 2017

A219452 First position of a plateau of length n in the product-of-exponents function A005361.

Original entry on oeis.org

2, 4, 843, 74848, 671345, 8870024

Offset: 2

R. J. Mathar, Nov 20 2012

nonn

The smallest index k such that A005361(k+1) = A005361(k+2) = ... = A005361(k+n).

a(8) is conjectured to be 1770019255373287038727484868192109228823.

J.-M. de Koninck, F. Luca, The product of exponents in the factorization of consecutive integers, Mathematika 55 (2009) 59-65

Cf. A005361.

A253288 Each term a(n) satisfies four properties: 1, divisible by all prime factors of n; 2, divisible by only the prime factors of n; 3, not equal to any of the terms a(1), a(2), ... a(n-1); 4, smallest number satisfying 1-3 if A005361(n) is even, or second smallest number satisfying 1-3 if A005361(n) is odd.

Original entry on oeis.org

1, 4, 9, 2, 25, 12, 49, 16, 3, 20, 121, 6, 169, 28, 45, 8, 289, 18, 361, 10, 63, 44, 529, 36, 5, 52, 81, 14, 841, 60, 961, 64, 99, 68, 175, 24, 1369, 76, 117, 50, 1681, 84, 1849, 22, 15, 92, 2209, 48, 7, 40, 153, 26, 2809, 72, 275, 98, 171, 116, 3481, 30, 3721, 124, 21, 32

Offset: 1

N. J. A. Sloane, Dec 29 2014

nonn

This sequence is permutation of the positive integers.
The prime p occurs at n = p^2.
Multiples of a number x have density 1/x.
Conjecture: this permutation of positive integers is self-inverse. Compare with A358971. The principal distinction between this sequence and A358971 is that fixed points aside from A358971(1) = 1 are explicitly ruled out in the latter. - Michael De Vlieger, Dec 10 2022

Brad Klee, Posting to Sequence Fans Mailing List, Dec 21, 2014.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n) <= 12000, n = 1..2^10 showing primes in red, other prime powers (in A246547) in gold, squarefree composites (in A120944) in green, numbers neither squarefree nor prime power (in A120706) in blue and magenta. The terms in magenta are products of composite prime powers (in A286708).
Michael De Vlieger, Log log scatterplot of a(n) <= 2^14, n = 1..2^14, showing a(n) such that rad(n) = 6 in red, and A358971(n) such that rad(n) = 6 in blue for comparison. This is an example of a self-inverse relation among terms a(n) in A003586.
Michael De Vlieger, Log log scatterplot of a(n) <= 80000, n = 1..2^14, showing a(n) in tiny black points if a(n) = A358971(n), else a(n) in red, and A358971(n) in blue.
Index entries for sequences that are permutations of the natural numbers

Cf. A005361 (Product of exponents of prime factorization of n), A358971.

A253288div := proc(a,n)
    local npr,d,apr ;
    npr := numtheory[factorset](n) ;
    for d in npr do
        if modp(a,d) <> 0 then
            return false;
        end if;
    end do:
    apr := numtheory[factorset](a) ;
    if apr minus npr = {} then
        true;
    else
        false;
    end if;
end proc:
A253288 := proc(n)
    option remember;
    local a,i,prev,act,ev ;
    if n =1 then
        1;
    else
        act := 1 ;
        if type(A005361(n),'even') then
            ev := true;
        else
            ev := false;
        end if;
        for a from 1 do
            prev := false;
            for i from 1 to n-1 do
                if procname(i) = a then
                    prev := true;
                    break;
                end if;
            end do:
            if not prev then
                if A253288div(a,n) then
                    if ev or act > 1 then
                        return a;
                    else
                        act := act+1 ;
                    end if;
                end if;
            end if;
        end do:
    end if;
end proc:
seq(A253288(n),n=1..80) ; # R. J. Mathar, Jan 22 2015

nn = 1000; c[] = False; q[] = 1; f[n_] := f[n] = Map[Times @@ # &, Transpose@ FactorInteger[n]]; a[1] = 1; c[1] = True; u = 2; Do[Which[PrimeQ[n], k = n^2, PrimeQ@ Sqrt[n], k = Sqrt[n], SquareFreeQ[n], k = First@ f[n]; m = q[k]; While[Nand[! c[k m], k m != n, Divisible[k, First@ f[m]]], m++]; While[Nor[c[q[k] k], Divisible[k, First@ f[q[k]]]], q[k]++]; k *= m, True, t = 0; Set[{k, s}, {First[#], 1 + Boole@ OddQ@ Last[#]} &[f[n]]]; m = q[k]; Until[t == s, If[m > q[k], m++]; While[Nand[! c[k m], Divisible[k, First@f[m]]], m++]; t++]; If[s == 1, While[Nor[c[q[k] k], Divisible[k, First@ f[q[k]]]], q[k]++]]; k *= m]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Dec 10 2022 *)

Terms beyond 361 from R. J. Mathar, Jan 22 2015

cp's OEIS Frontend

A005934 Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization (A005361).

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A322327 a(n) = A005361(n) * A034444(n).

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A284001 a(n) = A005361(A283477(n)).

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A322328 a(n) = A005361(n) * 4^A001221(n) for n > 0.

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A353500 Numbers that are the smallest number with product of prime exponents k for some k. Sorted positions of first appearances in A005361, unsorted version A085629.

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A085651 Index of the first of two successive 2's in A005361.

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PARI

PARI

A052306 Product of exponents of prime factorization of n by prime signature: A005361(A025487).

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